Advertisement

Mathematische Annalen

, Volume 374, Issue 1–2, pp 653–680 | Cite as

On the Fourier spectrum of functions on Boolean cubes

  • Andreas Defant
  • Mieczysław Mastyło
  • Antonio PérezEmail author
Article
  • 81 Downloads

Abstract

Let f be a real-valued function of degree d defined on the n-dimensional Boolean cube \(\{ \pm 1\}^{n}\), and \(f(x) = \sum _{S \subset \{1,\ldots ,n\}} \widehat{f}(S) \prod _{k \in S} x_k\) its Fourier-Walsh expansion. The main result states that there is an absolute constant \(C >0\) such that the \(\ell _{2d/(d+1)}\)-sum of the Fourier coefficients of \(f:\{ \pm 1\}^{n} \longrightarrow [-1,1]\) is bounded by \(C^{\sqrt{d \log d}}\). It was recently proved that a similar result holds for complex-valued polynomials on the n-dimensional polytorus \(\mathbb {T}^n\), but that in contrast to this, a replacement of the n-dimensional torus \(\mathbb {T}^n\) by the n-dimensional cube \([-1, 1]^n\) leads to a substantially weaker estimate. This in the Boolean case forces us to invent novel techniques which differ from the ones used in the complex or real case. We indicate how our result is linked with several questions in quantum information theory.

Keywords

Boolean functions polynomial inequalities Fourier analysis on groups 

Mathematics Subject Classification

06E30 47A30 81P45 

References

  1. 1.
    Aaronson, S., Ambainis, A.: The need for structure in quantum speedups. Theory Comput. 10, 133–166 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bayart, F., Pellegrino, D., Seaoane-Sepúlveda, J.B.: The Bohr radius of the \(n\)-dimensional polydisk is equivalent to \(\sqrt{(\log {n})/n}\). Adv. Math. 264, 726–746 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Blei, R.: Analysis in integer and fractional dimensions, Cambridge Studies in Advanced Mathematics 71 (2001)Google Scholar
  4. 4.
    Bohnenblust, H.F., Hille, E.: On the absolute convergence of Dirichlet series. Ann. Math. 32(3), 600–622 (1931)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Borwein, P., Erdélyi, T.: Polynomials and polynomial inequalities. Graduate Texts in Mathematics, 161. Springer, New York (1995)CrossRefGoogle Scholar
  6. 6.
    Campos, J.R., Jiménez Rodríguez, P., Muñoz-Fernández, G.A., Pellegrino, D., Seoane-Sepúlveda, J.B.: On the real polynomial Bohnenblust–Hille inequality. Linear Algebra Appl. 465, 391–400 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Defant, A., Frerick, L., Ortega-Cerdá, J., Ounaäies, M., Seip, K.: The Bohnenblust-Hille inequality for homogeneous polynomials is hypercontractive. Ann. Math (2) 174(1), 485–497 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Defant, A., Mastyło, M.: Bohnenblust-Hille inequalities for Lorentz spaces via interpolation. Anal. PDE 9(5), 1235–1258 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Defant, A., Mastyło, M., Pérez, A.: Bohr’s phenomenon for functions on the Boolean cube; J. Funct. Anal. (2018).  https://doi.org/10.1016/j.jfa.2018.05.009
  10. 10.
    De Wolf, R.: A brief introduction to Fourier analysis on the Boolean cube. Theory Comput Graduate Surv 1, 1–20 (2008)Google Scholar
  11. 11.
    Dinur, I., Friedgut, E., Kindler, G., O’Donnell, R.: On Fourier tails of bounded functions over the discrete cube. Israel J. Math. 160, 389–412 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Harris, L.A.: Bounds on the derivatives of holomorphic functions of vectors. Analyse fonctionnelle et applications (Comptes Rendus Colloq. Analyse, Inst. Mat., Univ. Federal Rio de Janeiro, Rio de Janeiro, 1972), 145–163. Actualités Aci. Indust., No. 1367, Hermann, Paris (1975)Google Scholar
  13. 13.
    Klimek, M.: Metrics associated with extremal plurisubharmonic functions. Proc. Am. Math. Soc. 123(9), 2763–2770 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kwapień, S.: Decoupling inequalities for polynomial chaos. Ann. Probab. 15(3), 1062–1071 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Littlewood, J.E.: On bounded bilinear forms in an infinite number of variables. Quart. J. Math. 1, 164–174 (1930)CrossRefzbMATHGoogle Scholar
  16. 16.
    Montanaro, A.: Some applications of hypercontractive inequalities in quantum information theory. J. Math. Phys. 53, 122–206 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Natanson, I.P.: Constructive function theory, vol. 1. Frederick Ungar, New York (1964)zbMATHGoogle Scholar
  18. 18.
    O’Donnell, R: Some topics in analysis of Boolean functions. In: Proc. 40th ACM Symp. Theory Comput., 569–578. ACM, New York (2008)Google Scholar
  19. 19.
    O’Donnell, R.: Analysis of Boolean functions. Cambridge University Press, New York (2014)CrossRefzbMATHGoogle Scholar
  20. 20.
    O’Donnell, R., Zhao, Y.: Polynomial bounds for decoupling with applications, 31st Conference on Computational Complexity, Art. No. 24, LIPIcs. Leibniz Int. Proc. Inform., 50, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern (2016)Google Scholar
  21. 21.
    ODonnell, R., Saks, M., Schramm, O., Servedio, R.: Every decision tree has an influential variable, In Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, 31–39 (2005)Google Scholar
  22. 22.
    Visser, C.: A generalization of Tchebychef’s inequality to polynomials in more than one variable. Indagationes Math. 8, 310–311 (1946)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut für MathematikCarl von Ossietzky UniversitätOldenburgGermany
  2. 2.Faculty of Mathematics and Computer SciencesAdam Mickiewicz University in PoznańPoznańPoland
  3. 3.Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Campus Cantoblanco UAMMadridSpain

Personalised recommendations